Does Fermat’s Last Theorem hold in ℚ(√2)? No, in fact it can be shown that if n>3 then the equation Xⁿ+Yⁿ=Zⁿ has no non-trivial solutions in ℚ(√2,√3) (therefore not even in ℚ(√2) however for n=3 we have that:
(18+17√2)³+(18-17√2)³=42³
Source: https://eprints.whiterose.ac.uk/194494/14/Fermat.pdf, https://math.stackexchange.com/questions/4438180/does-fermats-last-theorem-imply-sqrt2-not-in-mathbbq
Marco Damele 
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